CH. XVII] COMPARISON OF RELATED FORMS 1027 



We are apt to think of mathematical definitions as too strict and 

 rigid for common use, but their rigour is combined with all but 

 endless freedom. The precise definition of an ellipse introduces us 

 to all the elHpses in the world; the definition of a "conic section" 

 enlarges our concept, and a "curve of higher order" all the toore 

 extends our range of freedom*. By means of these large limitations, 

 by this controlled and regulated freedom, we reach through mathe- 

 matical analysis to mathematical synthesis. We discover homologies 

 or identities which were not obvious before, and which our descrip- 

 tions obscured rather than revealed : as for instance, when we learn 

 that, however we hold our chain, or however we fixe our bullet, the 

 contour of the one or the path of the other is always mathematically 

 homologous. 



Once more, and this is the greatest gain of all, we pass quickly and 

 easily from the mathematical concept of form in its statical aspect to 

 form in its dynanncal relations: we rise from the conception of 

 form to an understanding of the forces which gave rise to it; and 

 in the representation of form and in the comparison of kindred 

 forms, we see in the one case a diagram of forces in equihbrium, 

 and in the other case we discern the magnitude and the direction 

 of the forces which have sufficed to convert the one form into the 

 other. Here, since a change of material form is only effected by 

 the movement of matter |, we have once again the support of the 

 schoolman's and the philosopher's axiom, Ignorato motu, ignoratur 

 Natural' 



* So said Gustav Theodor Fechner, the author of Fechner's Law, a hundred 

 years ago. (Ueber die mathematische Behandlung organischer Gestalten und 

 Processe, Berichte d. k. sacks. Gesellsch., Maih.-phys. CI., Leipzig, 1849, pp. 50-64.) 

 Fechner's treatment is more purely mathematical and less physical in its scope and 

 bearing than ours, and his paper is but a short one, but the conclusions to which 

 he is led differ Uttle from our own. Let me quote a single sentence which, together 

 with its context, runs precisely on Aie lines which we have followed in this book : 

 "So ist also die mathematische Bestimmbarkeit im Gebiete des Organischen ganz 

 eben so gut vorhanden als in dem des Unorganischen, und in letzterem eben solchen 

 oder aquivalenten Beschrankungen unterworfen als in ersterem ; und nur sofern die 

 unorganischen Formen und das unorganische Geschehen sich einer einfacheren 

 Gesetzlichkeit mehr nahern als die organischen, kann die Approximation im 

 unorganischen Gebiet leichter und weiter getrieben werden als im organischen. 

 Dies ware der ganze, sonach rein relative, Unterschied." Here, in a nutshell, is 

 the gist of the whole matter. 



t "We can move matter, that is all we can do to it" (Oliver Lodge). 



